Syllogisms: the Forms of Logical Arguments
We evaluate arguments in logic with respect to two qualities: validity and soundness. The statements that we assume to be true or can establish to be true using good reasons for knowing stuff are called the premises of an argument, and the statements that we derive/infer from those premises are called conclusions. We say that an argument is valid if its form guarantees a true conclusion whenever the premises are true; we say that it is sound if it also has true premises, thus producing a true conclusion. Logical argument usually follows one or more of three acknowledged forms (called syllogisms) or their variants as listed below.
Modus Ponens
P Þ Q If a person has freckles, then he has Celtic blood.
P Liam has freckles.
\Q Therefore, Liam has Celtic blood.
Modus Tollens
P Þ Q If OU goes to a bowl game, then they have a winning season.
Ø Q OU cannot have a winning season.
\Ø P Therefore, OU will not go to a bowl game.
Disjunctive Syllogism
P Ú Q Either I will go to the game or I will attend the lecture.
Ø Q I will not go to the game.
\ P Therefore, I will attend the lecture.
Hypothetical Syllogism
P Þ Q If the horseshoe is missing a nail, the horse will falter.
Q Þ R If the horse falters, then the champion will fail.
R Þ S If the champion fails, then the battle is lost.
\ P Þ S If the horseshoe is missing a nail, then the battle is lost.
There is also a method of arguing indirectly called Reductio Ad Absurdum which is essentially a four step process.
1) Temporarily assume the negation of the proposition that you wish to prove.
2) Reason logically until that assumption produces a contradiction of a given or known fact such as a definition or law.
3) Declare the contradiction to be the result of the temporary assumption.
4) Assert the negation of the temporary assumption (which is the proposition that you wished to prove).
This is the form of argument most often used in alibis.
Example: Gertrude finds herself pregnant and accuses Horace of being the father of the child. Horace knows that he is not the guilty party and uses a reductio ad absurdum argument in his defense: “Suppose I were the father of the unborn child. Then DNA testing would show a high correlation between the baby’s chromosome pattern and my chromosome pattern. However, DNA testing does not show a high correlation between the patterns. This contradicts the assumption that I am the father. Therefore, I cannot be the father of Gertrude’s child.”